Theorem isoml 40012

Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
isoml.b	⊢ 𝐵 = (Base‘𝐾)
isoml.l	⊢ ≤ = (le‘𝐾)
isoml.j	⊢ ∨ = (join‘𝐾)
isoml.m	⊢ ∧ = (meet‘𝐾)
isoml.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
isoml	⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦

Allowed substitution hints:   ∨ (𝑥,𝑦)   ≤ (𝑥,𝑦)   ∧ (𝑥,𝑦)   ⊥ (𝑥,𝑦)

Proof of Theorem isoml

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . 4 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isoml.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2816	. . 3 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		isoml.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2816	. . . . . 6 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 5120	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
8		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		isoml.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2816	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11		eqidd 2764	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑥 = 𝑥)
12		fveq2 6881	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13		isoml.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
14	12, 13	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
15		eqidd 2764	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑦 = 𝑦)
16		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17		isoml.o	. . . . . . . . . 10 ⊢ ⊥ = (oc‘𝐾)
18	16, 17	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = ⊥ )
19	18	fveq1d 6883	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( ⊥ ‘𝑥))
20	14, 15, 19	oveq123d 7431	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑥)))
21	10, 11, 20	oveq123d 7431	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))
22	21	eqeq2d 2774	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
23	7, 22	imbi12d 347	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
24	3, 23	raleqbidv 3338	. . 3 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
25	3, 24	raleqbidv 3338	. 2 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
26		df-oml 39953	. 2 ⊢ OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
27	25, 26	elrab2 3654	1 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  lecple 17312  occoc 17313  joincjn 18362  meetcmee 18363  OLcol 39948  OMLcoml 39949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-oml 39953

This theorem is referenced by:  isomliN  40013  omlol  40014  omllaw  40017